We give conditions for non-conservative dynamics in reversible maps with transverse and non-transverse homoclinic orbits.
The paper is devoted to an investigation of the genus of an orientable closed surface M2 which admits A-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller Λr with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if M2 is a torus or a sphere, then M2 admits such an endomorphism. We also show that, if Ω is a basic set with a uniquely defined unstable bundle of the endomorphism f : M2 → M2 of a closed orientable surface M2 and f is not a diffeomorphism, then Ω cannot be a Cantor type expanding attractor. At last, we prove that, if f : M2 → M2 is an A-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type Ωr with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of Ωr is regular, then M2 is a two-dimensional torus T2 or a two-dimensional sphere S2.
In this paper, we consider a class of orientation-preserving Morse–Smale diffeomorphisms defined on an orientable surface. The papers by Bezdenezhnykh and Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of a heteroclinic intersection. However, such graphs generally do not admit polynomial discriminating algorithms. This article proposes a new approach to the classification of these cascades. For this, each diffeomorphism under consideration is associated with a graph that allows the construction of an effective algorithm for determining whether graphs are isomorphic. We also identified a class of admissible graphs, each isomorphism class of which can be realized by a diffeomorphism of a surface with an orientable heteroclinic. The results obtained are directly related to the realization problem of homotopy classes of homeomorphisms on closed orientable surfaces. In particular, they give an approach to constructing a representative in each homotopy class of homeomorphisms of algebraically finite type according to the Nielsen classification, which is an open problem today.
In this paper, we consider regular topological flows on closed n-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse –Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse –Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold.
In this article we study the plasma motion in the transitional layer of a coronal loop randomly driven at one of its footpoints in the thin-tube and thin-boundary-layer (TTTB) approximation. We introduce the average of the square of a random function with respect to time. This average can be considered as the square of the oscillation amplitude of this quantity. Then we calculate the oscillation amplitudes of the radial and azimuthal plasma displacement as well as the perturbation of the magnetic pressure. We find that the amplitudes of the plasma radial displacement and the magnetic-pressure perturbation do not change across the transitional layer. The amplitude of the plasma radial displacement is of the same order as the driver amplitude. The amplitude of the magnetic-pressure perturbation is of the order of the driver amplitude times the ratio of the loop radius to the loop length squared. The amplitude of the plasma azimuthal displacement is of the order of the driver amplitude times Re1/6, where Re is the Reynolds number. It has a peak at the position in the transitional layer where the local Alfvén frequency coincides with the fundamental frequency of the loop kink oscillation. The ratio of the amplitude near this position and far from it is of the order of , where is the ratio of thickness of the transitional layer to the loop radius. We calculate the dependence of the plasma azimuthal displacement on the radial distance in the transitional layer in a particular case where the density profile in this layer is linear
The study of deformations of Lie algebras is related to the problem of classification of simple Lie algebras over fields of small characteristics. The classification of finite-dimensional simple Lie algebras over algebraically closed fields of characteristic p > 3 is completed. Over fields of characteristic 2, a large number of examples of Lie algebras are constructed that do not fit into previously known schemes. Description of deformations of classical Lie algebras gives new examples of simple Lie algebras and gives a possibility to describe known examples as deformations of classical Lie algebras. In this paper, we describe global deformations of Lie algebras of the type Dn and their quotient algebras Dn by the center in the case of a field of characteristic 2.
This paper deals with one-dimensional factor maps for the geometric model of Lorenz-type attractors in the form of two-parameter family of Lorenz maps on the interval 𝐼=[−1,1]I=[−1,1] given by 𝑇𝑐,𝜈(𝑥)=(−1+𝑐⋅|𝑥|𝜈)⋅𝑠𝑖𝑔𝑛(𝑥)Tc,ν(x)=(−1+c⋅|x|ν)⋅sign(x). This is the normal form for splitting the homoclinic loop with additional degeneracy in flows with symmetry that have a saddle equilibrium with a one-dimensional unstable manifold. Due to L. P. Shilnikov’ results, such a bifurcation (under certain conditions) corresponds to the birth of the Lorenz attractor. We indicate those regions in the parameter plane where the topological entropy depends monotonically on the parameter 𝑐c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.
This paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any AA-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of a chaotic one-dimensional canonically embedded surface attractor and repeller.
We consider Cr-diffeomorphisms (1≤r≤+∞) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily Cr-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order rr, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that Cr-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.
We study the geometry of the bifurcation diagrams of the families of vector fields in the plane. Countable number of pairwise non-equivalent germs of bifurcation diagrams in the two-parameter families is constructed. Previously, this effect was discovered for three parameters only. Our example is related to so-called saddle node (SN)–SN families: unfoldings of vector fields with one saddle-node singular point and one saddle-node cycle. We prove structural stability of this family. By the way, the tools that may be helpful in the proof of structural stability of other generic two-parameter families are developed. One of these tools is the embedding theorem for saddle-node families depending on the parameter. It is proved at the end of the paper.
The role of various long-wave approximations in the description of the wave field and bottom pressure caused by surface waves, and their relation to evolution equations are being considered. In the framework of the linear theory, these approximations are being tested on the well-known exact solution for the wave spectral amplitudes and pressure variations. The famous Whitham, Korteweg–de Vries (KdV) and Benjamin–Bona–Mahony (BBM) equations have been used as evolutionary equations. It has been shown that if the wave is long, though steep enough, the BBM approximation gives better results than the KdV approximation, and they are quite close to the exact results. The same applies to the description of rogue waves, though formed from smooth relatively long waves, are often short and steep, then they may be invisible in variations of the bottom pressure. Another advantage of the BBM approximation for calculating the bottom pressure is the ability to analyze noisy series without preliminary filtering, which is necessary when using the KdV approximation.
Meteotsunamis are long waves generated by displacement of a water body due to atmospheric pressure disturbances that have similar spatial and temporal characteristics to landslide tsunamis. NAMI DANCE that solves the nonlinear shallow water equations is a widely used numerical model to simulate tsunami waves generated by seismic origin. Several validation studies showed that it is highly capable of representing the generation, propagation and nearshore amplification processes of tsunami waves, including inundation at complex topography and basin resonance. The new module of NAMI DANCE that uses the atmospheric pressure and wind forcing as the other inputs to simulate meteotsunami events is developed. In this paper, the analytical solution for the generation of ocean waves due to the propagating atmospheric pressure disturbance is obtained. The new version of the code called NAMI DANCE SUITE is validated by comparing its results with those from analytical solutions on the flat bathymetry. It is also shown that the governing equations for long wave generation by atmospheric pressure disturbances in narrow bays and channels can be written similar to the 1D case studied for tsunami generation and how it is integrated into the numerical model. The analytical solution of the linear shallow water model is defined, and results are compared with numerical solutions. A rectangular shaped flat bathymetry is used as the test domain to model the generation and propagation of ocean waves and the development of Proudman resonance due to moving atmospheric pressure disturbances. The simulation results with different ratios of pressure speed to ocean wave speed (Froude numbers) considering sub-critical, critical and super-critical conditions are presented. Fairly well agreements between analytical solutions and numerical solutions are obtained. Additionally, basins with triangular (lateral) and stepwise shelf (longitudinal) cross sections on different slopes are tested. The amplitudes of generated waves at different time steps in each simulation are presented with discussions considering the channel characteristics. These simulations present the capability of NAMI DANCE SUITE to model the effects of bathymetric conditions such as shelf slope and local bathymetry on wave amplification due to moving atmospheric pressure disturbances.
We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble the well-known discrete Lorenz-like attractors and those which are similar to the discrete Shilnikov attractors. We describe and illustrate the scenarios of occurrence of chaotic attractors of both types.
To study stationary periodic weakly vortical waves on water (the Gouyon waves), the method of the modified Lagrangian variables is suggested. The wave vorticity Ω is specified as a series in the small steepness parameter ε in the form: Ω =\sum( ε^n · Ω_n (b)), where Ω_n are arbitrary functions of the vertical Lagrangian coordinate b. Earlier Gouyon (1958) studied waves in linear and quadratic approximations. The present paper provides a complete problem solution in a cubic approximation. Explicit expressions are obtained for the coordinates of the liquid particle trajectories and pressure. The quadratic correction to the wave velocity is determined. The comparison with the Stokes and Gerstner waves is made. The class of quadratically vortical Gouyon waves is distinguished, for which Ω_1 = 0. On the basis of the general dispersion theory method for waves of this class, the nonlinear Schrödinger equation is written. The criterion, when the equation is focusing and rogue Gouyon waves can be formed, is indicated.
On December 22, 2018, a destructive tsunami related to the phenomena caused by the volcanic eruption of Gunung Anak Krakatau (GAK) was generated following a partial collapse of the volcano that caused serious damage and killed more than 400 people. This recent event challenged the traditional understanding of tsunami hazard, warning and response mechanisms and raised the topic of volcanic tsunami hazard. The complex mechanism of this tsunamigenic volcano collapse still needs further investigation as Anak Krakatau is one of the potentially tsunamigenic volcanoes in the world. This study investigates the possible source mechanisms of this phenomenon and their contribution to explaining the observed sea level disturbances by considering the impacts of this destructive event. We configured a flank collapse scenario with a volume of 0.25 km3 as a combination of submarine and subaerial mass movement as the possible source scenarios to the December 22, 2018 Sunda Strait tsunami. A two-layer model is applied to simulate the tsunami generation by these landslides up to 420 s. The tsunami propagation and inundation are then simulated by NAMI DANCE model in GPU environment. The simulation results suggest that this scenario seems capable of generating such a tsunami observed along the coast of Sunda Strait. However, the contribution of the possible submarine mass movements in the close area between GAK and the surrounding islands either to this event or potential tsunami threat in the region is still questionable. We employed a bathymetric dataset through pre- and post-event analyses, which demonstrate submarine slope failures in the southwestern proximity of GAK. Hence, additional two scenarios of elliptical landslide sources on the slopes of bathymetry change area (could be triggered by seismic motion/volcanic eruption) are considered, searching for the possible effects of the tsunami that might be generated by these submarine landslides. The study may also provide some perspective for potential tsunami generation by combined sources and help to elucidate the extent of volcanic tsunami hazard in the region due to potential future eruptions of Gunung Anak Krakatau.
Bifurcations that occur in a small neighborhood of a polycycle of a planar vector field are called semilocal. We prove that even semilocal bifurcations of hyperbolic polycycles may have numeric invariants of topological classification
The present paper gives a partial answer to Smale’s question which diagrams can correspond to (A,B)-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class G of (A,B)-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class G realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from G with isomorphic labeled Smale diagrams which are not ambiently Ω-conjugated are constructed. Moreover, a subset G∗ ⊂ G of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient Ω-conjugacy is singled out.
The existence problem for attractors of foliations with transverse linear connection is investigated. In general foliations with transverse linear connection do not admit attractors. A conditions that implies the existence of a global attractor which is a minimal set, is specified. An application to transversely similar pseudo-Riemannian foliations is obtained. The global structure of transversely similar Riemannian foliations is described. Different examples are constructed.
It is a well known fact that any smooth manifold admits a Morse function, whereas the problem of existence of a Morse function for a topological manifold stated by Marston Morse in 1959 is still open. In the present paper we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with a finite hyperbolic chain recurrent set. We construct this function as a Lyapunov function whose set of the critical points coincides with the chain recurrent set of the flow.
We consider a class of gradient-like flows on three-dimensional closedmanifolds whose attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the ambient manifold is Seifert.